What is Manning's equation?

Manning's equation gives the mean velocity of uniform open-channel flow: V = (1/n) R^(2/3) S^(1/2), where n is Manning's roughness coefficient, R is the hydraulic radius (area/wetted perimeter), and S is the channel bed slope. Discharge Q = A V.

What is hydraulic radius?

Hydraulic radius R = A/P is the ratio of the flow cross-sectional area A to the wetted perimeter P (the length of channel boundary in contact with water). It represents the flow's efficiency - a larger R means less boundary friction per unit area.

What is the most economical channel section?

The most economical (or most efficient) section carries the maximum discharge for a given area, which happens when the wetted perimeter is minimum and hydraulic radius is maximum. For a rectangular channel this occurs when the width equals twice the depth (b = 2y), giving R = y/2.

Manning's Equation and Open Channel Flow — Design with Worked Examples

Rivers, canals, drains and sewers flowing partly full are open channels — driven by gravity, with a free surface at atmospheric pressure. The workhorse design equation is Manning's equation, used worldwide for sizing channels and estimating discharge. This article explains it with worked examples.

Open Channel Flow Basics

Unlike pipe flow (driven by pressure), open channel flow is driven by the bed slope and has a free surface. Uniform flow occurs when depth and velocity are constant along the channel — the gravity component balances the boundary friction.

Manning's Equation

V = (1/n) · R^(2/3) · S^(1/2)        Q = A · V

Symbol	Meaning
V	Mean velocity (m/s)
n	Manning's roughness coefficient (dimensionless)
R = A/P	Hydraulic radius (m)
S	Bed slope (m/m)
A	Flow area; P = wetted perimeter

Typical Manning's n Values

Surface	Manning's n
Smooth concrete / cement plaster	0.012 – 0.014
Brickwork / ordinary concrete	0.015 – 0.017
Earthen channel, clean	0.018 – 0.025
Natural stream with weeds/stones	0.030 – 0.050

Worked Example 1 — Discharge in a Rectangular Channel

A rectangular concrete channel is 3 m wide, flowing 1 m deep. Bed slope S = 0.001, n = 0.015. Find the velocity and discharge.

Area A = b·y = 3 × 1 = 3 m²
Wetted perimeter P = b + 2y = 3 + 2 = 5 m
Hydraulic radius R = A/P = 3/5 = 0.6 m
V = (1/0.015) × 0.6^(2/3) × 0.001^(1/2) = 66.67 × 0.7114 × 0.03162 = 1.50 m/s
Q = A·V = 3 × 1.50 = 4.5 m³/s

Worked Example 2 — Slope Required for a Target Discharge

The same channel must carry 6 m³/s at 1 m depth. Find the bed slope required.

Required V = Q/A = 6/3 = 2.0 m/s
From Manning: S = (V·n / R^(2/3))² = (2.0 × 0.015 / 0.7114)² = (0.04218)² = 0.00178 (≈ 1 in 562)

Most Economical Rectangular Section

For a given area, the most efficient rectangular channel minimises the wetted perimeter. Setting dP/dy = 0 with A = b·y constant gives:

b = 2y     and     R = y/2

This carries the maximum discharge for the least lining — a key result for economical canal design. (For a trapezoidal channel, the most economical section is a half-hexagon.)

Worked Example 3 — Most Economical Section

Design the most economical rectangular concrete channel (n = 0.015) to carry 5 m³/s on a slope of 0.0009.

Most economical: b = 2y, R = y/2, A = b·y = 2y²
Q = A·V = 2y² × (1/n)(y/2)^(2/3) S^(1/2)
5 = 2y² × (1/0.015)(y/2)^(2/3)(0.0009)^(1/2) = 2y² × 66.67 × (y/2)^(2/3) × 0.03
5 = 4.0 y² (y/2)^(2/3) → solving, y ≈ 1.07 m, b = 2y ≈ 2.14 m

Applications

Irrigation canal and field-channel design.
Storm-water drains, road-side drains and culvert sizing.
Sewer design (running partly full) and natural-stream flow estimation.

Common Mistakes

Using the full perimeter (including the free surface) as wetted perimeter — the water surface is not a wetted boundary.
Confusing hydraulic radius R with hydraulic depth or pipe radius.
Inconsistent units — Manning's n above is for SI; US customary units carry a 1.49 factor.